We also provide the topological classification of essential shifts. To do this, it is more natural to work with a
certain finite index subgroup of \(\operatorname{Map} (\Sigma )\) . Let \(E\)  be the end space of \(\Sigma \) . Mann-Rafi [1]}
defined a partial order on \(E\)  measuring the local complexity. Let \(\operatorname{FMap} (\Sigma )\)  be the subgroup of
\(\operatorname{Map} (\Sigma )\)  that fixes the set of isolated maximal points in the end space \(E\) 
(see Section REF  for the definition). In the setting of CB (but infinite) generated
groups, it is not always true that a finite index subgroup \(H\)  of \(G\)  is quasi-isometric to \(G\) . However, we show:
